\documentclass[]{article}
\setlength{\pdfpagewidth}{8.5in} \setlength{\pdfpageheight}{11in}

\usepackage{natbib}
           \setlength{\bibhang}{.3in}
           \setlength{\bibsep}{.05in plus0.05in minus0.05in}
           \makeatletter
           \renewcommand{\bibsection}{
           \begin{center}
           \section*{\refname\@mkboth{\MakeUppercase{\refname}}
               {\MakeUppercase{\refname}}}
           \end{center}
           }


	\bibliographystyle{jf}
	
	\begin{document}
	 
	 
	 

     \begin{table}[htbp] 
  \caption{\bf Correlations:  local-PCA vs IPCA }
\medskip
\setlength{\tabcolsep}{0pt}
\begin{tabular}{lccccccc|} 
 $ \scriptsize{(N,T):}$ & (15,5)  & (15,25) &  (100,5) & (100,25) & (1,000,5) & (1,000,25) \\  \hline 
& \multicolumn{5}{c}{\scriptsize{\bf Experiment (A): Correct Model - Stationary Instruments} } & \\
  & \scriptsize{ (local-PCA ,IPCA) } & \scriptsize{ (local-PCA ,IPCA)  } & \scriptsize{(local-PCA ,IPCA) } & \scriptsize{(local-PCA ,IPCA) } & \scriptsize{ (local-PCA ,IPCA)  } & \scriptsize{ (local-PCA ,IPCA) }  \\ 
 \scriptsize{ correlation ${\bf F }$:} &                       (0.61 , 0.71) &    (0.31,0.71) &  (0.63,0.93) &  (0.32,0.93) &    (0.58,0.96) & (0.31,0.94)  \\ 
 \scriptsize{ correlation ${\bf \Lambda }_{t-1} $:} & (0.30 , 0.75) &  (0.25, 0.75) &  (0.16,0.94) &   (0.13,0.93) &    (0.14,0.92) & (0.09,0.89) \\  \hline
   & \multicolumn{5}{c}{\scriptsize{\bf Experiment (B): Misspecified Model - Stationary Instruments} } & \\
  & \scriptsize{ (local-PCA ,IPCA) } & \scriptsize{ (local-PCA ,IPCA)  } & \scriptsize{(local-PCA ,IPCA) } & \scriptsize{(local-PCA ,IPCA) } & \scriptsize{ (local-PCA ,IPCA)  } & \scriptsize{ (local-PCA ,IPCA) }  \\ 
 \scriptsize{ correlation ${\bf F }$:}                      & (0.61 , 0.24) & (0.31,0.24) &  (0.63,0.48) &      (0.32,0.45) &      (0.58,0.46) &  (0.31,0.55)  \\
 \scriptsize{ correlation ${\bf \Lambda }_{t-1} $:} & (0.30 , 0.23) &  (0.25,0.26) &   (0.16,0.26) &      (0.13,0.26) &    (0.14,0.21) & (0.09,0.26) \\   \hline 
   & \multicolumn{5}{c}{\scriptsize{\bf Experiment (C): Correct Model - Random Walk Instruments} } & \\
  & \scriptsize{ (local-PCA ,IPCA) } & \scriptsize{ (local-PCA ,IPCA)  } & \scriptsize{(local-PCA ,IPCA) } & \scriptsize{(local-PCA ,IPCA) } & \scriptsize{ (local-PCA ,IPCA)  } & \scriptsize{ (local-PCA ,IPCA) }  \\ 
 \scriptsize{ correlation ${\bf F }$:} &                      (0.58 , 0.22) &  (0.42,0.23) & (0.64,0.19) &      (0.38,0.21) &  (0.59,0.18) & ( 0.32,0.18) \\
 \scriptsize{ correlation ${\bf \Lambda }_{t-1} $:} & (0.25 , 0.25) &  (0.22,0.25)   &  (0.09,0.27) &    (0.09,0.28) &  (0.03,0.27) & (0.03,0.23) \\  \hline
    \end{tabular}
    \parbox{7.25in}{\footnotesize{This  table presents the absolute value of the correlations (averaged across $1,000$ Monte Carlo iterations) between the true and estimated first risk factor  $ { F }_{1t}$ (first row of each panel) and between the true and estimated  first risk exposure  $ {\lambda }_{1it-1} $ (second row of each panel), for various combinations of $(N,T)$. The data generating process is (OA38) when the $10$-dimensional vector of instruments ${\bf z }_{it-1}$ is a stationary VAR (Experiment (A))  and a multivariate random walk (Experiment (C)), respectively. Experiment (B) differs from (A) because the estimated model is misspecified (one instrument instead of $10$ instruments).
}}
\label{FIG15MCOAdue}
\end{table}




   \begin{table}[htbp] 
  \caption{\bf   Total-$R^2$:  local-PCA vs IPCA }
\medskip
\setlength{\tabcolsep}{0pt}
 \begin{tabular}{lccccccc|} 
 $ \scriptsize{(N,T):}$ & (15,5)  & (15,25) &  (100,5) & (100,25) & (1,000,5) & (1,000,25) \\  \hline 
 & \multicolumn{5}{c}{\scriptsize{\bf Experiment (A): Correct Model - Stationary Instruments} } & \\
  & \scriptsize{ (local-PCA,IPCA) } & \scriptsize{ (local-PCA,IPCA)  } & \scriptsize{(local-PCA,IPCA) } & \scriptsize{(local-PCA,IPCA)} & \scriptsize{(local-PCA,IPCA)} & \scriptsize{(local-PCA,IPCA) }  \\ 
 \scriptsize{ total-$R^2$ (in-sample):} &      (0.84,0.34) &    (0.51,0.33) &  (0.74,0.17) &  (0.34,0.16) & (0.75,0.15) & (0.35,0.14)  \\ 
 \scriptsize{ total-$R^2$ (out-of-sample):} & (0.25,0.27) &  (0.29,0.27) &  (0.11,0.16) & (0.17,0.15) & (0.15,0.14) &
  (0.23,0.14)  \\  \hline
  & \multicolumn{5}{c}{\scriptsize{\bf Experiment (B): Misspecified Model - Stationary Instruments} } & \\
 & \scriptsize{ (local-PCA,IPCA) } & \scriptsize{ (local-PCA,IPCA)  } & \scriptsize{(local-PCA,IPCA) } & \scriptsize{(local-PCA,IPCA) } & \scriptsize{ (local-PCA,IPCA)  } & \scriptsize{ (local-PCA,IPCA) }  
  \\ 
 \scriptsize{  total-$R^2$ (in-sample):} &    (0.84,0.03) &  (0.51,0.03) & (0.74,0.02) & (0.34,0.02) &  (0.75,0.01)  & 
 (0.35,0.01) \\
 \scriptsize{ total-$R^2$ (out-of-sample):} & (0.25,0.03) &  (0.29,0.03) &   (0.11,0.01) &  (0.17,0.01) &  (0.15,0.01) & (0.23,0.01) \\   \hline 
    & \multicolumn{5}{c}{\scriptsize{\bf Experiment (C): Correct Model - Random Walk Instruments} } & \\
  & \scriptsize{ (local-PCA,IPCA) } & \scriptsize{ (local-PCA,IPCA)  } & \scriptsize{(local-PCA,IPCA) } & \scriptsize{(local-PCA,IPCA) } & \scriptsize{ (local-PCA,IPCA)  } & \scriptsize{ (local-PCA,IPCA) }  
  \\ 
 \scriptsize{ total-$R^2$(in-sample):} & (0.84,0.29) &  (0.55,0.29) & (0.76,0.05) & (0.43,0.06) &  (0.75,0.02) &
   (0.46,0.02)\\
 \scriptsize{ total-$R^2$ (out-of-sample):} & (0.27,0.21) &  (0.34,0.21)   &  (0.14,0.04) & (0.25,0.04) &  (0.18,0.02) & (0.31,0.02) \\  \hline
    \end{tabular}
    \parbox{7.25in}{\footnotesize{This  table presents the 
  total-$R^2$ metric of  Kelly et al (2019)[Eq. (15)]  (averaged across $1,000$ Monte Carlo iterations) defined as $ R^2 \equiv 1 - \sum_{i=1}^N \sum_{s=1}^{T_0} ( x_{is} -  \boldsymbol{\tilde{ \lambda}}_{is-1} \tilde{\bf f}_s )^2/ \sum_{i=1}^N \sum_{s=1}^{T_0} x_{is}^2 $, where $\boldsymbol{\tilde{ \lambda}}_{is-1} ,\tilde{\bf f}_s  $ denote the  estimated risk exposures and factors by either local-PCA or IPCA, evaluated in-sample and out-of-sample,  for various combinations of $(N,T)$. The data generating process is (OA38)
  when the $10$-dimensional vector of instruments ${\bf z }_{it-1}$ is a  stationary VAR (Experiment (A))  and a multivariate random walk (Experiment (C)), respectively. Experiment (B) differs from (A) because the estimated model is misspecified (one instrument instead of $10$ instruments).
}}
\label{FIG16MCOAdue}
\end{table}






 \begin{table}[htbp] 
 
   \caption{\bf Pricing Ability:  local-PCA vs  IPCA }
 \medskip
\setlength{\tabcolsep}{0pt}
begin{tabular}{lccccccc|}   
 $ \scriptsize{(N,T):}$ & (15,5)  & (15,25) &  (100,5) & (100,25) & (1,000,5) & (1,000,25) \\  \hline 
& \multicolumn{5}{c}{\scriptsize{\bf Experiment (A): Correct Model - Stationary Instruments} } & \\
  & \scriptsize{ (IPCA/local-PCA) } & \scriptsize{ (IPCA/local-PCA)  } & \scriptsize{(IPCA/local-PCA) } & \scriptsize{(IPCA/local-PCA)} & \scriptsize{(IPCA/local-PCA)} & \scriptsize{(IPCA/local-PCA) }  \\ 
 \scriptsize{
 In-Sample:} &    8.761   &  3.473 &   7.963 &    3.640  &  11.055 &   4.479 \\
 \scriptsize{ 
 Out-of-Sample:} &  1.519 &  2.830 &    1.849 &    2.870 &    2.006 &    3.332  \\  \hline
  & \multicolumn{5}{c}{\scriptsize{\bf Experiment (B): Misspecified Model - Stationary Instruments} } & \\
  & \scriptsize{ (IPCA/local-PCA) } & \scriptsize{ (IPCA/local-PCA)  } & \scriptsize{(IPCA/local-PCA) } & \scriptsize{(IPCA/local-PCA)} & \scriptsize{(IPCA/local-PCA)} & \scriptsize{(IPCA/local-PCA) }  \\ 
  \scriptsize{  
 In-Sample:}   &     14.599 &   6.123 &    9.263 &    3.586 &  12.307 &   5.311 \\
 \scriptsize{ 
 Out-of-Sample:} &     2.204  &   4.231 &    2.133 &    2.794 &   2.224 &    3.988 \\  \hline
   & \multicolumn{5}{c}{\scriptsize{\bf Experiment (C): Correct Model - Random Walk Instruments} } & \\
  & \scriptsize{ (IPCA/local-PCA) } & \scriptsize{ (IPCA/local-PCA)  } & \scriptsize{(IPCA/local-PCA) } & \scriptsize{(IPCA/local-PCA)} & \scriptsize{(IPCA/local-PCA)} & \scriptsize{(IPCA/local-PCA) }  \\
 \scriptsize{  
 In-Sample:} &  8.977 &   3.606  &    8.534 &    3.356 &   10.983 &    4.739  \\
 \scriptsize{ 
 Out-of-Sample:} &    1.565 &    3.036 &   1.897 &    2.849 &   2.016 &    3.493 \\ \hline
    \end{tabular}
    \parbox{7.0in}{\footnotesize{This  table presents ratios of the pricing ability criterion $N^{-1}\sum_{i=1}^N \tilde{\delta}_{i}^2  $  corresponding to the IPCA  (numerator) and  local-PCA (denominator),   averaged across $1,000$ Monte Carlo iterations,  and evaluated in-sample and out-of-sample,  for various combinations of $(N,T)$. The data generating process is (15) where the  vector ${\bf z }_{it-1}$ is a $10$-sized stationary VAR (Experiment (A)) and a multivariate random walk (Experiment (C)), respectively. Experiment (B) differs from (A) because the IPCA-estimated model is misspecified (one instrument instead of the $10$ required instruments).
}}
\label{Table1OAdue}
\end{table}

	
\bibliography{Biblio_PCA}


\end{document}
